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Non-vanishing of $$L$$-functions and their derivatives. (English) Zbl 0745.11033
Automorphic forms and analytic number theory, Proc. Conf., Montréal/Can. 1989, 89-113 (1990).
[For the entire collection see Zbl 0728.00008.]
The paper in detail discusses the main result of M. R. Murty and V. K. Murty [Ann. Math., II. Ser. 133, No. 3, 447-475 (1991; see the preceding review)] and its implications for modular elliptic curves $$E/\mathbb{Q}$$ and also gives an outline of the proofs. In the last section, the author discusses some heuristics on the order of vanishing of $$L(E,s)$$ at $$s=1$$ and also analogous questions for Artin $$L$$-functions and Dirichlet $$L$$-functions.

##### MSC:
 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11F67 Special values of automorphic $$L$$-series, periods of automorphic forms, cohomology, modular symbols 11G05 Elliptic curves over global fields 14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) 14H25 Arithmetic ground fields for curves 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 11M41 Other Dirichlet series and zeta functions